88
Chapter 5. The Derivative
Chapter 6
Sequences and Series of
Functions
6.1
Discussion: Branching Processes
6.2
Uniform Convergence of a Sequence
of Functions
Exercise 6.2.1. (a) By dividing the nume
5.4. A Continuous Nowhere-Dierentiable Function
85
Turning our attention to the dierence quotient, we get
g(xm ) g(0)
=
xm 0
Pm
m
X
1/2m
=
1 = m + 1.
1/2m
n=0
n=0
Because this quantity increases witho
94
Chapter 6. Sequences and Series of Functions
and choose N2 so that
|gn gm | < /2
for all m, n N2 .
Letting N = maxcfw_N1 , N2 we see that
|(fn + gn ) (fm + gm )| |fn fm | + |gn gm |
<
+ =
2 2
for
100
Chapter 6. Sequences and Series of Functions
Because this holds for all m N , we can set m = n 1 to get that
|fn (x)| <
whenever n > N .
This proves gn ! 0 uniformly.
Exercise 6.4.2. The key idea
5.3. The Mean Value Theorem
79
closer and closer to zero. To see this explicitly, set tn = 1/(2n) and xn = 0.
Then observe that |xn tn | ! 0 while
g2 (xn ) g2 (tn )
0
g2 (tn ) = |tn sin(1/tn ) + cos
103
6.5. Power Series
P
Exercise 6.5.6. Assume
an xn converges pointwise on the compact set K.
Because K is compact, there exist points x0 , x1 2 K satisfying
x0 x x1
for all x 2 K.
At this point we n
6.2. Uniform Convergence of a Sequence of Functions
91
Exercise 6.2.5. (a) Taking the limit for each fixed value of x we find that fn (x)
converges pointwise to
1 if x 6= 0
f (x) =
0 if x = 0.
Each of
106
Chapter 6. Sequences and Series of Functions
Exercise 6.6.3. The key idea is to take the derivative of each side of equation
(2) using a term-by-term approach for the series on the right (this is
6.3. Uniform Convergence and Dierentiation
97
Letting N = maxcfw_N1 , N2 , . . . , Nm produces the desired N .
Note that if the set cfw_r1 , r2 , . . . , rm were infinite then N would be the maximum
82
Chapter 5. The Derivative
() For the other direction we use the Mean Value Theorem. Here we are
assuming f 0 (x) 0 on (a, b) and we are asked to prove that f is increasing.
Given x < y, it follows